Excitation Control Design for Stochastic Power Systems with Input Delay Based on Nonlinear Hamiltonian System Theory

This paper presents H ∞ excitation control design problem for power systems with input time delay and disturbances by using nonlinear Hamiltonian system theory. The impact of time delays introduced by remote signal transmission and processing in wide-area measurement system (WAMS) is well considered. Meanwhile, the systems under investigation are disturbed by random fluctuation. First, under prefeedback technique, the power systems are described as a nonlinear Hamiltonian system. Then the


Introduction
Time delay always exists in power systems control area.It is often ignored when controller is mainly applied in local systems where the communication time delay is very small compared to the system time constants (see, e.g., [1,2] and the references therein).Due to the further study of phase measurement unit (PMU) and WAMS, coordinated stability control has got a lot of attention.It uses remote measuring information given by WAMS/PMU.Unlike the small delay in local control, the time delay in wide-area power systems can vary from tens to several hundred milliseconds or more.Since that the large time delay will go against the stability of the system and reduce the performance of the system, so it is very necessary to consider the influence of it on the power system stability analysis and controller design.Besides, the generators are interfered with speed regulation, fluctuation of load, mechanical torsional vibration, the changes of damping coefficients, and so on in the transient process.These random fluctuations can be regarded as a kind of random process [3].However, the application of the Itô differential formula will lead to the appearance of gravitation and the Hessian term.What is more, the stochastic disturbance (Wiener process) will cause no definition of the system states' derivative [4].Therefore, stochastic and delay factors increase the difficulties of the analysis and synthesis [5].Some results, which took signal transmission time delays or stochasticity in power systems into account, have been obtained.Reference [6] presented a free-weighting matrix method based on linear control design approach for the wide-area robust damping controller associated with flexible alternating current transmission system device to improve the dynamical performance of the large-scale power systems.Reference [7] proposed a delay-independent decentralized coordinated robust approach to design excitation controller in terms of  ∞ optimization method incorporating linear matrix inequality (LMI) technique.Considering the nonlinear effects of randomized torsional oscillation on the excitation regulation dynamic process of a generator rotor and exploiting Monte-Carlo principle and numerical methods, the algorithms and workflow of the proposed excitation control system's transient stability analysis approach were presented in [3].Reference [8] presented a stochastic cost model and a solution technique for optimal scheduling of the generators in a wind integrated power system considering the demand and wind generation uncertainties.
Based on the linearization at steady state operating point, lots of the techniques are by far achieved and applied to controller design in power systems.These techniques have some disadvantages, such as ignoring some nonlinearities of the system and just expressing the partial structures of the system.What is more, the designed controllers are generally relatively complicated and not very easy to realize online operation.Therefore, some nonlinear methods should be worked out to achieve good control performance for the power systems in consideration of time-delay, stochastics, and disturbances.In recent years, energy-based Lyapunov function method has obtained numerous attention, and remarkable achievements have been reached with this method in the analysis and synthesis of nonlinear systems, as well as in the power systems (see, e.g., [9][10][11][12][13] and the references therein).The method can thoroughly take advantage of the internal structural properties of the systems and make the control design relatively simple.An important step in using energy-based control strategy is to transform the system into a dissipative Hamiltonian system formulation.This kind of system, proposed by [14], has great benefits for that its Hamilton function can be used as the sum of potential energy (excluding gravitational potential energy) and kinetic energy in physical systems and also can be taken as a Lyapunov function (see, e.g., [11,[15][16][17][18]).Using the energy-based Hamilton function method, [11] investigated the adaptive  ∞ excitation control of multimachine power systems with disturbances.Simulations show that the control strategy proposed in [11] was more effective than some other control schemes.Considering the impact of time delays in acquisition and transmission of key signals in power systems, [19] deals with the  ∞ excitation control problem of machine power system with time-delay and disturbances.
The purpose of this paper is to present a suitable controller structure for the stochastic power systems with input delay and disturbances using the nonlinear Hamiltonian system theory in order to weaken the impact of stochasticity and delay on the control performance of the power systems.Firstly, the prefeedback with delay method is to be used to describe the system as a dissipative Hamiltonian system formulation.Next, based on the obtained new system formulation, we will deal with the  ∞ control problem by using Newton-Leibniz formula, a few properties of norm and matrices.The main results will be proposed for the Hamiltonian system and the power system as well.Finally, we will test and verify the obtained results in this paper by an example of a two-machine power system with delay, stochasticity, and disturbances.
The rest of the paper is organized as follows.Section 2 provides the problem formulation, nonlinear Hamilton realization and some preliminaries.Section 3 gives the main results.Analysis of the achieved results by a two-machine power system example and the conclusion are given in Sections 4 and 5, respectively.
Notations.Throughout the paper the superscript "" stands for matrix transposition.R denotes the set of real numbers, R + the set of all nonnegative real numbers, R  the dimensional Euclidean space, and R × the real matrices with dimension  × .Diag{⋅ ⋅ ⋅ } stands for diagonal matrix in which the diagonal elements are the elements in {⋅ ⋅ ⋅ }; ‖ ⋅ ‖ stands for either the Euclidean vector norm or the induced matrix 2-norm.For any symmetric matrices  and ,  ≥  (resp.,  > ) means that the matrix - is positive semidefinite (resp., positive definite).tr[] denotes the trace for square matrix . max () ( min ()) denotes the maximum (minimum) of eigenvalue of a real symmetric matrix .
) denotes the family of all F 0 -measurable bounded C([−, 0]; R  )-valued random variables  = {() :  ∈ [−, 0]}.C  denotes the set of all functions with continuous th partial derivatives; C 2,1 is the family of all functions which are C 2 in the first argument and C 1 in the second argument; C 2,1 (R  × [−, ∞); R + ) stands for the family of all nonnegative functions (, ) on R  × [−, ∞) which are C 2 in  and C 1 in .What is more, for the sake of simplicity, throughout the paper, we denote / by ∇.

Problem Formulation and Nonlinear Hamilton Realization
Consider the following -machine power systems, each generator of which is described by a third-order dynamic model (see [1,20]): where is the power angle of the th generator (radians),   is the rotor speed of the th generator (rad/s),  0 = 2 0 ,    is the -axis internal transient voltage of the th generator (per unit),   is the -axis transient reactance (per unit),    is the -axis transient reactance of the th generator (per unit),   is the voltage of the field circuit of the th generator, the control input (per unit),   is the inertia coefficient of the th generator (s),   is the damping constant (per unit),  0 is the field circuit time constant (s),   is the mechanical power, assumed to be constant (per unit),   is the active electrical power (per unit),  1 and  2 are bounded disturbances,   is the -axis current (per unit),   is the internal voltage (per unit),   =   +   is the admittance of line - (per unit), and   =   +   is the self-admittance of bus  (per unit).
There are signal transmission delays and random process in the modern power systems.The delays in the measuring data exist in such case that the exciter inputs are taken from remote buses.And assume that all the feedback widearea signals have the time delay .Meanwhile, the generator torque can be regarded as a kind of random process because of random fluctuation in transient process, such as speed regulation, fluctuation of load, mechanical torsional vibration, and the changes of damping coefficients.Moreover, considering the imaginary control input is   which feeds back both the local measurement information and the widearea measurement signals, so the power system (1) should be modeled into differential-algebraic equations with time delay and stochasticity as follows: where  is random disturbance intensity and () is a zero-mean Wiener process on a probability space (Ω, F, P) relative to an increasing family (F  ) >0 of  algebras (F  ) >0 ⊂ F; here Ω is the samples space, F is  algebra of subsets of the sample space, and  is the probability measure on F.Moreover, we assume {()} = 0, {[()] 2 } = , where  is the expectation operator.Assume that ( (0)  ,  0 ,  (0)  ),  = 1, 2, . . ., , are the preassigned operating points of system (3). Setting . ., , then system (3) can be rewritten as follows: Inspired by [11], we introduce a prefeedback control law: where the first term is to make system (4) have a Hamilton structure, the second and third terms are to guarantee the operating point of the system unchanged,   ( − ) is the new reference input, and   and   are undetermined constants.
To make the operating point of the system invariant,   and   have to satisfy and   =  0 which is spelled out in [11]; what is more, this reference provides a kind of choice of   and   .Furthermore, (5) can be rewritten as Let ] T , then system (4) can be expressed as a dissipative Hamiltonian system as follows: where ) , ∇  (  ) is the gradient of the Hamilton function   (  ), which satisfies   (0) = 0,  = 1, 2, . . ., .
Owing to each individual subsystem having the crossvariables, this structure does not provide the overall system a Hamilton structure.Thus, we need to find out a common Hamilton function for the  generators, which is regarded as the total energy of the whole system. Let where  = [ T 1 , . . .,  T  ] T .By using relation   =   , we can verify that which imply that () is the common Hamilton function for the  generators.Furthermore, () ∈ C 2 holds obviously.Setting then system (8) can be rewritten as follows: where Obviously,  is a skew-symmetric matrix, and  is a positive semidefinite matrix.In addition, we can choose  =  T  2 ∇() and  =  T 1 ∇() as the output and the penalty signal, respectively, where  is a full column rank weighting matrix.Definition 1.The stochastic time delay Hamiltonian system (13) is said to be robustly asymptotically stable in mean square, if there exists a controller ( − ) such that lim where  0 is the preassigned equilibrium and () is the solution of system (13) at time  under initial condition.
Consider the following cost function: Then  ∞ control objective of system ( 13) is to find a feedback controller: such that for given  > 0 and at the same time the closed-loop system is asymptotically stable when  = 0.
We conclude this section by recalling some auxiliary results to be used in this paper.
If there exists a function such that for some constant  > 0 and any  ≥ 0, L ≤  (1 +  ( () , ) +  ( ( − ) ,  − )) , where the differential operator L is defined as then there exists a unique solution on [−, ∞) for any initial data Lemma 3.For any given matrices  ∈ R × and  ∈ R × , there holds Proof.This proof can be achieved by using the properties of matrix's trace.

Hamiltonian System.
The  ∞ controller is given below for the stochastic Hamiltonian system (13) with input delay.
Next step we prove the closed-loop system where system (13) under the control law ( 24) is asymptotically stable in mean square when  = 0.
When  = 0, from (35), we can easily get that Set Furthermore, owing to (A4) holding, there is In addition, because of {()} = 0, we further get Hence, one has From condition (A3), one has Set Multiplying  − 1  to the two sides of inequality (44) yields Integrating inequality (48) from  0 to , we have Due to  1 < 0, there is lim According to Definition 1, we can conclude that system (13) under the control law (24) is robustly asymptotically stable in mean square with respect to  0 .This completes the proof.

𝑁-Machine
Power System.In this subsection, we consider the -machine power system (3).
First, we can verify that Choose the preassigned equilibrium satisfying and ∇ T () = 0; that is Meanwhile, we assume that there exist positive constants An  ∞ controller for system (3) is given in the following theorem.

Theorem 7. Consider power system (3). If
into consideration, then we can prove the result using the similar method in the proof of Theorem 4, where   ≥ 0,  = 1, 2, . . .,  are the weighting constants.

Illustrative Examples
To show the effectiveness of the proposed control strategy, we give a two-machine power system as shown in Figure 1.
The generators  1 ,  2 are assumed to be connected to distant power systems and disturbed by random fluctuation.In simulating, a temporary short-circuit fault occurs at point  (see Figure 1) during the time 0.5 sec∼1 sec.The system parameters used in this simulation are given in Table 1.
We will test the effectiveness of the proposed control configuration at two different time delays  = 0.5 s and  = 0.05 s.The initial condition is (63) Simulations with the above initial condition and the delay  = 0.5 s and  = 0.05 s are given in Figures 2-4 and Figures 5-7, separately.Through Figures 2-7, we can see that the states of the system converge to the equilibrium ( 1 (0),  1 (0),   1 (0),  2 (0),  2 (0),   2 (0)) =   [1.2 1 2 1.2 1 2] eventually.Obviously, under the delayed feedback controller by using the proposed method, the robustness of the closed-loop system is guaranteed.It is also seen that the controller possesses insensitivity in regard to the types of time delay and stochastic disturbances.

Conclusion
This paper studied the  ∞ excitation controller design problem of a class of stochastic power systems with time-delay and disturbances.In the design process, we used the prefeedback technique, Newton-Leibniz formula, and a few properties of norm.Besides, we obtain these results by nonlinear Hamilton function approach due to the special structural properties of the Hamiltonian systems.We also give a two-machine power system simulation and it shows that the results achieved in this paper are practicable in analyzing the  ∞ excitation control problem of stochastic power system in consideration of time-delay and disturbances.